For the time being, the cheapest way to get to Livorno from Pisa Airport, and vice versa, is by using local transport:
By bus and train - there is currently work underway at Pisa airport to provide a continuous "Pisa Mover" train connection with Pisa Central railway station. This work is due to be completed in December 2015.
At the moment (Jan 2015)there is a bus connection from the airport to Pisa Centrale station (the Bus Pisa Mover). This runs every 10 minutes from 6am to midnight daily and takes approximately 8 minutes. It costs €1.30 each way. From Pisa Centrale there are regular direct trains to Livorno (€2.50 each way on Regional trains). Please see the Trenitalia website for the full timetable.
For information about competitive car hire from Pisa Airport, click here
When all else fails, a taxi from the airport to Livorno costs around 50 euros, depending on time of day and amount of luggage.
Alternatively, to book a private transfer from Pisa Airport, see the recommended companies on this page